As described on Wikipedia at http://en.wikipedia.org/wiki/Vibration, which is incorporated by reference herein, vibration refers to mechanical oscillations about an equilibrium point. The oscillations may be periodic such as the motion of a pendulum or random such as the movement of a tire on a gravel road. Vibration is occasionally “desirable”. For example the motion of a tuning fork, the reed in a woodwind instrument or harmonica, or the cone of a loudspeaker is desirable vibration, necessary for the correct functioning of the various devices. More often, vibration is undesirable, wasting energy and creating unwanted sound—noise. For example, the vibrational motions of engines, electric motors, or any mechanical device in operation are typically unwanted. Such vibrations can be caused by imbalances in the rotating parts, uneven friction, the meshing of gear teeth, etc.
There are two classes of vibration. Free vibration occurs when a mechanical system is set off with an initial input and then allowed to vibrate freely. Examples of this type of vibration are pulling a child back on a swing and then letting go or hitting a tuning fork and letting it ring. The mechanical system will then vibrate at one or more of its “natural frequencies” and damp down to zero. Forced vibration is when an alternating force or motion is applied to a mechanical system. Examples of this type of vibration include a shaking washing machine due to an imbalance, transportation vibration (caused by truck engine, springs, road, etc.), or the vibration of a building during an earthquake. In forced vibration the frequency of the vibration is the frequency of the force or motion applied, with order of magnitude being dependent on the actual mechanical system.
The fundamentals of vibration analysis can be understood by studying the simple mass-spring-damper model. Indeed, even a complex structure such as an automobile body can be modeled as a “summation” of simple mass-spring-damper models. The mass-spring-damper model is an example of a simple harmonic oscillator. The mathematics used to describe its behavior is identical to other simple harmonic oscillators such as the RLC circuit.
To start the investigation of the mass-spring-damper one can assume the damping is negligible and that there is no external force applied to the mass (i.e. free vibration). The force applied to the mass by a spring is proportional to the amount the spring is stretched “x”. The proportionality constant, k, is the stiffness of the spring and has units of force/distance (e.g. lbf/in or N/m). The negative sign indicates that the force is always opposing the motion of the mass attached to it.Fs=−kx. 
The force generated by the mass is proportional to the acceleration of the mass as given by Newton's second law of motion.
      ∑    F    =      ma    =                  m        ⁢                  x          ¨                    =              m        ⁢                                                            ⅆ                2                            ⁢              x                                      ⅆ                              t                2                                              .                    
The sum of the forces on the mass then generates the ordinary differential equation:m{umlaut over (x)}+kx=0.
If we assume that we start the system to vibrate by stretching the spring by the distance of A and letting go, the solution to the above equation that describes the motion of mass is:x(t)=A cos(2πfnt).
This solution says that it will oscillate with simple harmonic motion that has an amplitude of A and a frequency of fn. The number fn is one of the most important quantities in vibration analysis and is called the undamped natural frequency. For the simple mass-spring system, fn is defined as:
      f    n    =            1              2        ⁢        π              ⁢                            k          m                    .      
Angular frequency ω (ω=2πf) with the units of radians per second is often used in equations because it simplifies the equations, but is normally converted to “standard” frequency (units of Hz or equivalently cycles per second) when stating the frequency of a system.
If you know the mass and stiffness of the system you can determine the frequency at which the system will vibrate once it is set in motion by an initial disturbance using the above stated formula. Every vibrating system has one or more natural frequencies that it will vibrate at once it is disturbed. This simple relation can be used to understand in general what will happen to a more complex system once we add mass or stiffness. For example, the above formula explains why when a car or truck is fully loaded the suspension will feel “softer” than unloaded because the mass has increased and therefore reduced the natural frequency of the system while the spring stiffness remains constant.
Vibrational motion could be understood in terms of conservation of energy. In the above example we have extended the spring by a value of x and therefore have stored some potential energy (½kx2) in the spring. Once we let go of the spring, the spring tries to return to its un-stretched state (which is the minimum potential energy state) and in the process accelerates the mass. At the point where the spring has reached its un-stretched state all the potential energy that we supplied by stretching it has been transformed into kinetic energy (½mv2). The mass then begins to decelerate because it is now compressing the spring and in the process transferring the kinetic energy back to its potential. Thus oscillation of the spring amounts to the transferring back and forth of the kinetic energy into potential energy.
In the simple model the mass will continue to oscillate forever at the same magnitude, but in areal system there is always something called damping that dissipates the energy, eventually bringing it to rest.
A “viscous” damper can be added to the model that outputs a force that is proportional to the velocity of the mass. The damping is called viscous because it models the effects of an object within a fluid. The proportionality constant c is called the damping coefficient and has units of Force over velocity (lbf s/in or N s/m).
      F    d    =            -      cv        =                            -          c                ⁢                  x          .                    =                        -          c                ⁢                                            ⅆ              x                                      ⅆ              t                                .                    
By summing the forces on the mass we get the following ordinary differential equation:m{umlaut over (x)}+c{dot over (x)}+kx=0
The solution to this equation depends on the amount of damping. If the damping is small enough the system will still vibrate, but eventually, over time, will stop vibrating. This case is called underdamping—this case is of most interest in vibration analysis. If we increase the damping just to the point where the system no longer oscillates, we reach the point of critical damping (if the damping is increased past critical damping the system is called overdamped). The value that the damping coefficient needs to reach for critical damping in the mass spring damper model is:Cc=2√{square root over (km)}. 
To characterize the amount of damping in a system a ratio called the damping ratio (also known as damping factor and % critical damping) is used. This damping ratio is just a ratio of the actual damping over the amount of damping required to reach critical damping. The formula for the damping ratio (ζ) of the mass spring damper model is:
  ζ  =            c              2        ⁢                  km                      .  
For example, metal structures (e.g. airplane fuselage, engine crankshaft) will have damping factors less than 0.05 while automotive suspensions in the range of 0.2-0.3.
The solution to the underdamped system for the mass spring damper model is the following:x(t)=Xe−ζωnt cos(√{square root over (1−ζ2ωnt−φ))}, ωn=2πfn.
The value of X, the initial magnitude, and φ, the phase shift, are determined by the amount the spring is stretched. The formulas for these values can be found in the references.
The major points to note from the solution are the exponential term and the cosine function. The exponential term defines how quickly the system “damps” down—the larger the damping ratio, the quicker it damps to zero. The cosine function is the oscillating portion of the solution, but the frequency of the oscillations is different from the undamped case.
The frequency in this case is called the “damped natural frequency”,fd, and is related to the undamped natural frequency by the following formula:fd=√{square root over (1−ζ2)}fn.
The damped natural frequency is less than the undamped natural frequency, but for many practical cases the damping ratio is relatively small and hence the difference is negligible. Therefore the damped and undamped description are often dropped when stating the natural frequency (e.g. with 0.1 damping ratio, the damped natural frequency is only 1% less than the undamped).
We will now describe the behavior of the spring mass damper model when we add a harmonic force in the form below. A force of this type could, for example, be generated by a rotating imbalance.F=F0 cos(2πft)
If we again sum the forces on the mass we get the following ordinary differential equation:m{umlaut over (x)}+c{dot over (x)}+kx=F0 cos(2πft).
The steady state solution of this problem can be written as:x(t)=X cos(2πft−φ).
The result states that the mass will oscillate at the same frequency, f, of the applied force, but with a phase shift φ.
The amplitude of the vibration “X” is defined by the following formula.
  X  =                    F        0            k        ⁢                  1                                                            (                                  1                  -                                      r                    2                                                  )                            2                        +                                          (                                  2                  ⁢                  ζ                  ⁢                                                                          ⁢                  r                                )                            2                                          .      
Where “r” is defined as the ratio of the harmonic force frequency over the undamped natural frequency of the mass-spring-damper model.
  r  =            f              f        n              .  
The phase shift, φ, is defined by the following formula.
  ϕ  =            arctan      ⁡              (                              2            ⁢            ζ            ⁢                                                  ⁢            r                                1            -                          r              2                                      )              .  
The plots of the amplitude and phase functions provided in FIGS. 7A and 7B, which correspond to the “frequency response of the system”, present one of the most important features in forced vibration. In a lightly damped system when the forcing frequency nears the natural frequency (r≈1) the amplitude of the vibration can get extremely high. This phenomenon is called resonance (subsequently the natural frequency of a system is often referred to as the resonant frequency). In rotor bearing systems any rotational speed that excites a resonant frequency is referred to as a critical speed.
If resonance occurs in a mechanical system it can be very harmful—leading to eventual failure of the system. Consequently, one of the major reasons for vibration analysis is to predict when this type of resonance may occur and then to determine what steps to take to prevent it from occurring. As the amplitude plot shows, adding damping can significantly reduce the magnitude of the vibration. Also, the magnitude can be reduced if the natural frequency can be shifted away from the forcing frequency by changing the stiffness or mass of the system. If the system cannot be changed, perhaps the forcing frequency can be shifted (for example, changing the speed of the machine generating the force).
The following are some other points in regards to the forced vibration shown in the frequency response plots.                At a given frequency ratio, the amplitude of the vibration, X, is directly proportional to the amplitude of the force F0 (e.g. if you double the force, the vibration doubles)        With little or no damping, the vibration is in phase with the forcing frequency when the frequency ratio r<1 and 180 degrees out of phase when the frequency ratio r>1        When r<<1 the amplitude is just the deflection of the spring under the static force F0. This deflection is called the static deflection δst. Hence, when r<<1 the effects of the damper and the mass are minimal.        When r>>1 the amplitude of the vibration is actually less than the static deflection δst. In this region the force generated by the mass (F=ma) is dominating because the acceleration seen by the mass increases with the frequency. Since the deflection seen in the spring, X, is reduced in this region, the force transmitted by the spring (F=kx) to the base is reduced. Therefore the mass-spring-damper system is isolating the harmonic force from the mounting base—referred to as vibration isolation. Interestingly, more damping actually reduces the effects of vibration isolation when r>>1 because the damping force (F=cv) is also transmitted to the base.        
Resonance is simple to understand if you view the spring and mass as energy storage elements—with the mass storing kinetic energy and the spring storing potential energy. As discussed earlier, when the mass and spring have no external force acting on them they transfer energy back and forth at a rate equal to the natural frequency. In other words, if energy is to be efficiently pumped into both the mass and spring the energy source needs to feed the energy in at a rate equal to the natural frequency. Applying a force to the mass and spring is similar to pushing a child on swing, you need to push at the correct moment if you want the swing to get higher and higher. As in the case of the swing, the force applied does not necessarily have to be high to get large motions; the pushes just need to keep adding energy into the system.
The damper, instead of storing energy, dissipates energy. Since the damping force is proportional to the velocity, the more the motion, the more the damper dissipates the energy. Therefore a point will come when the energy dissipated by the damper will equal the energy being fed in by the force. At this point, the system has reached its maximum amplitude and will continue to vibrate at this level as long as the force applied stays the same. If no damping exists, there is nothing to dissipate the energy and therefore theoretically the motion will continue to grow on into infinity.
In a previous example only a simple harmonic force was applied to the model, but this can be extended considerably using two powerful mathematical tools. The first is the Fourier transform that takes a signal as a function of time (time domain) and breaks it down into its harmonic components as a function of frequency (frequency domain). For example, let us apply a force to the mass-spring-damper model that repeats the following cycle—a force equal to 1 newton for 0.5 second and then no force for 0.5 second. This type of force has the shape of a 1 Hz square wave.
The Fourier transform of the square wave generates a frequency spectrum that presents the magnitude of the harmonics that make up the square wave (the phase is also generated, but is typically of less concern and therefore is often not plotted). The Fourier transform can also be used to analyze non-periodic functions such as transients (e.g. impulses) and random functions. With the advent of the modern computer the Fourier transform is almost always computed using the Fast Fourier Transform (FFT) computer algorithm in combination with a window function.
In the case of our square wave force, the first component is actually a constant force of 0.5 newton and is represented by a value at “0” Hz in the frequency spectrum. The next component is a 1 Hz sine wave with an amplitude of 0.64. This is shown by the line at 1 Hz. The remaining components are at odd frequencies and it takes an infinite amount of sine waves to generate the perfect square wave. Hence, the Fourier transform allows you to interpret the force as a sum of sinusoidal forces being applied instead of a more “complex” force (e.g. a square wave).
In the previous section, the vibration solution was given for a single harmonic force, but the Fourier transform will in general give multiple harmonic forces. The second mathematical tool, “the principle of superposition”, allows you to sum the solutions from multiple forces if the system is linear. In the case of the spring-mass-damper model, the system is linear if the spring force is proportional to the displacement and the damping is proportional to the velocity over the range of motion of interest. Hence, the solution to the problem with a square wave is summing the predicted vibration from each one of the harmonic forces found in the frequency spectrum of the square wave.
We can view the solution of a vibration problem as an input/output relation—where the force is the input and the output is the vibration. If we represent the force and vibration in the frequency domain (magnitude and phase) we can write the following relation:
      X    ⁡          (      ω      )        =                              H          ⁡                      (            ω            )                          ·                  F          ⁡                      (            ω            )                              ⁢                          ⁢      or      ⁢                          ⁢              H        ⁡                  (          ω          )                      =                            X          ⁡                      (            ω            )                                    F          ⁡                      (            ω            )                              .      
H(ω) is called the frequency response function (also referred to as the transfer function, but not technically as accurate) and has both a magnitude and phase component (if represented as a complex number, a real and imaginary component). The magnitude of the frequency response function (FRF) was presented earlier for the mass-spring-damper system.
                                      H          ⁡                      (            ω            )                                      =                                                            X              ⁡                              (                ω                )                                                    F              ⁡                              (                ω                )                                                              =                              1            k                    ⁢                      1                                                                                (                                          1                      -                                              r                        2                                                              )                                    2                                +                                                      (                                          2                      ⁢                      ζ                      ⁢                                                                                          ⁢                      r                                        )                                    2                                                                          ;                  where        ⁢                                  ⁢        r            =                        f                      f            n                          =                  ω                      ω            n                                ,
The phase of the FRF was also presented earlier as:
      ∠    ⁢                  ⁢          H      ⁡              (        ω        )              =            arctan      ⁡              (                              2            ⁢            ζ            ⁢                                                  ⁢            r                                1            -                          r              2                                      )              .  
For example, let us calculate the FRF for a mass-spring-damper system with a mass of 1 kg, spring stiffness of 1.93 N/mm and a damping ratio of 0.1. The values of the spring and mass give a natural frequency of 7 Hz for this specific system. If we apply the 1 Hz square wave from earlier we can calculate the predicted vibration of the mass. FIG. 8 illustrates the resulting vibration. It happens in this example that the fourth harmonic of the square wave falls at 7 Hz. The frequency response of the mass-spring-damper therefore outputs a high 7 Hz vibration even though the input force had a relatively low 7 Hz harmonic. This example highlights that the resulting vibration is dependent on both the forcing function and the system that the force is applied to.
FIG. 8 also shows the time domain representation of the resulting vibration. This is done by performing an inverse Fourier Transform that converts frequency domain data to time domain. In practice, this is rarely done because the frequency spectrum provides all the necessary information.
The frequency response function (FRF) does not necessarily have to be calculated from the knowledge of the mass, damping, and stiffness of the system, but can be measured experimentally. For example, if you apply a known force and sweep the frequency and then measure the resulting vibration you can calculate the frequency response function and then characterize the system. This technique is used in the field of experimental modal analysis to determine the vibration characteristics of a structure.
As described on Wikipedia at http://en.wikipedia.org/wiki/Q_factor, which is incorporated by reference herein, vibration can also be described in relation to a quality (or Q) factor. In physics and engineering the quality factor or Q factor is a dimensionless parameter that describes how under-damped an oscillator or resonator is, or equivalently, characterizes a resonator's bandwidth relative to its center frequency. Higher Q indicates a lower rate of energy loss relative to the stored energy of the oscillator; the oscillations die out more slowly. A pendulum suspended from a high-quality bearing, oscillating in air, has a high Q, while a pendulum immersed in oil has a low one. Oscillators with high quality factors have low damping so that they ring longer.
Sinusoidally driven resonators having higher Q factors resonate with greater amplitudes (at the resonant frequency) but have a smaller range of frequencies around that frequency for which they resonate; the range of frequencies for which the oscillator resonates is called the bandwidth. Thus, a high Q tuned circuit in a radio receiver would be more difficult to tune, but would have more selectivity; it would do a better job of filtering out signals from other stations that lie nearby on the spectrum. High Q oscillators oscillate with a smaller range of frequencies and are more stable.
The quality factor of oscillators varies substantially from system to system. Systems for which damping is important (such as dampers keeping a door from slamming shut) have Q=½. Clocks, lasers, and other resonating systems that need either strong resonance or high frequency stability need high quality factors. Tuning forks have quality factors around Q=1000. The quality factor of atomic clocks, superconducting RF cavities used in accelerators, and some high-Q lasers can reach as high as 1011 and higher.
There are many alternative quantities used by physicists and engineers to describe how damped an oscillator is and that are closely related to the quality factor. Important examples include: the damping ratio, relative bandwidth, linewidth and bandwidth measured in octaves.
The concept of Q factor originated in electronic engineering, as a measure of the ‘quality’ desired in a good tuned circuit or other resonator.
As described at http://www.deicon.com/vib_categ, which is incorporated by reference herein, there are several basic approaches for controlling noise and vibration of a vibratory system. One common approach used to mitigate sound and vibration caused by acoustical/structural resonance, is adding damping to the acoustic plant and structure. Damping dissipates some of the sound/vibration energy by transforming it to heat.
Damping may or may not be effective depending on how close a disturbance frequency is to a resonant frequency being damped. In this case, passive or active cancellation solutions can be used to quiet a system. Passive sound/vibration cancellation is normally achieved by appending the oscillating system with a tuned absorber, e.g., Helmholtz resonators and quarter wave tubes (for sound) and dynamic absorbers (for vibration) with the natural frequency similar to the disturbing frequency.
Frequently, the goal of control is to prevent its transmission of sound/vibration to the surrounding. Such control schemes, known as ‘isolation’, are used extensively to isolate a noisy environment from a quiet one (in sound control), as well as machinery (industrial and marine), civil engineering structures (base isolation in building, bridges, etc.), and sensitive components from the foundation/base (in vibration control).
The most common passive isolation method is the use of sound barriers (in sound control) and mounting the vibrating structure/machine to the base via resilient elements, e.g., rubber, (in vibration control). Active isolation involves the use of actuators along with sensors and controllers (analog or digital) to create actuation with the goal of lowering the transmission of sound/vibration from one body to another. Although such isolation methods can be relatively successful isolating low frequency sound/vibration, high frequency sound/vibration is generally much more difficult to isolate. As such, there is a need for improved systems and methods for isolating vibration.
It is well known to apply mechanical impedance and mobility concepts in order to analyze vibratory systems to support corrective measures to control noise and vibration. As described in “A Guide to Mechanical Impedance and Structural Response Techniques', by H. P Olesen and R. B. Randall, Bruel & Kjaer Application Note No. 17-179, pp. 1-19. 1977, which is incorporated by reference herein, the basic concepts of mechanical impedance and mobility of forced vibratory systems were developed from electro-mechanical and electro-acoustic analogies in the 1920's. Generally, the mechanical impedance (F/v or Z) at a given point in a vibratory system is the ratio of the sinusoidal force applied to the system at that point to the velocity at the same point, where the dynamic mass (or apparent weight, F/a) is the ratio of the sinusoidal force to acceleration, and stiffness (F/d) is the ratio of the sinusoidal force to distance. Inversely, the mechanical admittance (or mobility, v/F) at a given point in a vibratory system is the ratio of the velocity applied to the system at that point to the sinusoidal force at the same point, the acceleration through force (a/F) is the ratio of acceleration to the sinusoidal force, and compliance (d/F) is the ratio of distance to the sinusoidal force. To solve vibrational problems, both a mechanical impedance may have to be measured and a narrow band frequency analysis may have to be performed to obtain detailed knowledge about the response ability of the structures (or objects) involved and of the actual responses or forces. From this information, the need or the possibility of corrective measures may be evaluated.
A lecture on simple acoustic filters by Dave L. Moulton at http://www.moultonworld. pwp.blueyonder.co.uk/Lecture9_page.htm, which is incorporated by reference herein, describes how common filters such as low pass, high pass, band pass and band stop filters can be realized in the acoustic domain. The lecture includes a discussion of a second order low pass filter circuit and includes a diagram of its frequency response, which is provided in FIG. 9. The diagram includes two frequency response curves that illustrate the effect of damping whereby under damping produces a resonance peak (or Laplace pole) and critical damping does not.
U.S. Pat. No. 4,912,727 to Wolfgang Schubert describes magnetic systems that transition from a mutual repel force to a mutual attract force depending on the distance of separation between pairs of magnetic components making up the magnetic systems. Schubert also describes the mutual repel force produced by the magnetic systems as providing a damping function. More specifically, Schubert teaches the mutual repel force produced by the magnetic system as being a means for decelerating the translational movement of the rails of a drawer guiding system and also being a means for accelerating the translational movement of the rails of the drawer guiding system.
The magnetic components of Shubert comprise complementary first portions each being two rows (or one dimensional arrays) of smaller alternating polarity individual magnets aligned with respect to a center line and complementary second portions each comprising a single larger magnet where the force curve produced by the two larger magnets must be “shallower” than the force curve produced by the rows of smaller alternating polarity individual magnets, but otherwise doesn't describe, teach, or suggest how or why the different shaped force curves are produced or can be affected.